| title |
author |
institute |
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Financial Modeling |
Keith A. Lewis |
KALX, LLC |
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\renewcommand\o[1]{\overline{#1}}
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In "A Simple Approach to the Valuation of Risky Streams"
Stephen Ross[@Ros1978] showed
If there are no arbitrage opportunities in the capital markets, then
there exists a (not generally unique) valuation operator, $L$.
As shown in the Unified Model,
Ross's linear valuation operators correspond to deflators: adapted,
positive, finitely-additive measures indexed by trading time.
Market instruments have prices and associated cash flows. Stocks have
dividends, bonds have coupons, futures have daily margin adjustments.
The price of a futures is always zero. A market model consist of
vector-values prices $(X_t)$ and cash flows $(C_t)$ indexed by market
instruments. Prices and cash flows depend only on information available
at time $t$. This is modeled by algebras of sets $\AA_t$
at each trading time $t$ and requiring prices
and cash flows to be measurable with respect to the algebras.
A model is arbitrage free if and only if there exist a deflators $(D_t)$ with
$$
\tag{1} X_t D_t = (X_u D_u + \sum_{t < s \le u}C_s D_s)|{\AA_t}
$$
where $|$ indicates restriction of measure. Recall a function
times a measure is a measure and the conditional expectation
${Y = E[X|\AA]}$ if and only if
$Y(P|{\AA}) = (XP)|_{\AA}$,
where $P$ is a probability measure.
If $(M_t)$ is s vector-valued martingale measure then
$$
X_t D_t = M_t - \sum_{s\le t}C_s D_s
$$
is an arbitrage-free model of prices and cash flows. For example,
the Black-Scholes/Merton model with no dividends corresponds to $X_t =
(r\exp(\rho t), s\exp(\rho t + \sigma B_t - \sigma^2 t/2))$, $C_t =
(0, 0)$ and ${D_t = \exp(-\rho t)P|_{\AA_t}}$ where $(B_t)$
is Brownian motion, $P$ is Wiener measure, and ${\AA_t}$ is the
standard filtration.
A trading strategy is a finite sequence $(\tau_j)$ of increasing
stopping times and vector-valued
$(\Gamma_j)$, depending on information available at time $\tau_j$,
indicating the number of shares purchased at time $\tau_j$.
Let $\Delta_t = \sum_{\tau_j < t} \Gamma_j = \sum_{s < t} \Gamma_s$ be the (settled) position at time $t$,
where $\Gamma_s = \Gamma_j$ if $s = \tau_j$ and is zero otherwise.
The amounts $A_t = \Delta_t\cdot C_t - \Gamma_t\cdot X_t$ show up in the
brokerage account during trading: you receive cash flows proportional to
existing positions and pay for trades just executed at the prevailing market prices.
The mark-to-market of the trading strategy at time $t$
is $V_t = (\Delta_t + \Gamma_t)\cdot X_t$. It the the value of unwinding
existing positions and the last trades at prevailing market prices.
A simple consequence of these definitions is
$$
\tag{2} V_t D_t = (V_u D_u + \sum_{t < s \le u}A_s D_s)|_{\AA_t}
$$
Note how value $V_t$ in (2) corresponds to prices $X_t$ in (1),
likewise amount $A_t$ to cash flows $C_t$.
Trading strategies create synthetic market instruments.
A derivative is a synthetic market instrument.
Its contract specifies a finite sequence of increasing stopping times
$\o{\tau}_j$ and amounts $\o{A}j$ paid at these times.
A European option has a single constant stopping time $\bar{\tau}$ with payoff
${\o{A} = \phi(X{\o{\tau}})}$ for some function $\phi$.
Suppose we could find a
a trading strategy $(\tau_j)$, $(\Gamma_j)$ with
$\sum_j \Gamma_j = 0$, $A_t = \o{A}_j$ when $t = \o{\tau}j$
and is zero (self-financing) otherwise. The condition ${\sum_j \Gamma_j = 0}$
requires the hedge to be eventually closed.
This is a perfect hedge and the value of the derivative at time $t$
would be determined by equation (2): ${V_t D_t = (\sum{\o{\tau}_j > t} \o{A}j D{\o{\tau}j})|{\AA_t}}$.
Perfect hedges do not exist in practice.
A fundamental problem in mathematical finance is how to hedge a
derivative when a perfect hedge does not exist.
A first attempt at a solution
is to assume a perfect hedge exists.
The initial hedge at $\tau_0 = 0$ can be computed from $V_0 = \Gamma_0\cdot X_0$
and ${V_0 D_0 = (\sum_{\o{\tau}j > t} \o{A}j D{\o{\tau}j})|{\AA_0}}$.
$$
\Gamma_0 D_0 = D{X_0}(\sum_{\o{\tau}j > 0} \o{A}j D{\o{\tau}j})|{\AA_0},
$$
where $D{X_0}$ is the Fréchet derivative. Just as in the B-S/M theory, the (putative) initial hedge is
the derivative of value with respect to current prices. Note that value can be
computed using only the deflators and the contract specified amounts.
For $\tau_1 = t > 0$ we have $V_t = (\Delta_t + \Gamma_t)\cdot X_t$ so
$$
(\Delta_t + \Gamma_t) D_t = D_{X_t}(\sum_{\o{\tau}_j > t} \o{A}j D{\o{\tau}j})|{\AA_0}.
$$
For $t > 0$ sufficiently small we have $\Delta_t = \Gamma_0$ so we can solve for $\Gamma_t$.
This procedure does not specify what value of $\tau_1$ to choose.
The Unified Model does not provide an
answer to when hedge, it only puts your nose directly in the problem of
when and how much to hedge.
The classical Black-Scholes/Merton theory gives the inapplicable answer that
you should trade "continuously".
If repurchase agreements exists then there is a canonical choice for the deflators.
A repurchase agreement with rate $f_t$, known at time $t$, has price 1 at
time $t$ and pays $\exp(f_t,dt)$ at time $t + dt$.
For any deflator $(D_t)$ equation (1) gives
${1 D_t = (\exp(f_t,dt)D_{t + dt})|{\AA_t} = \exp(f_t,dt)D{t + dt}|{\AA_t}}$.
For a deflator with $D{t + dt}$ known at time $t$ we have
${D_t = \exp(f_t,dt)D_{t + dt}}$.
The canonical deflator is ${D_t = \exp(-\int_0^t f_s,ds)D_0}$.
The repurchase rates $(f_t)$ are called the (continuously compounded) short rate process.
Every interest rate model is just a specific parameterization of this.
The deflators determine the prices of zero coupon bonds. If $D_t(u)$ is the price of
a zero coupon bond paying 1 unit at time $u$ then equation (1) implies
${D_t(u) D_t = D_u|_{\AA_t}}$ so the price is the Radon-Nikodym derivative.
The price of zero coupon bonds determine the value of all risk-free fixed income instruments.